Andreas Schadschneider Institute for Theoretical Physics University of Cologne Germany Modelling of Traffic Flow and Related Transport Problems

Overview Highway traffic Traffic on ant trails Pedestrian dynamics Intracellular transport basic phenomena modelling approaches theoretical analysis physics Topics: Aspects: General topic: Application of nonequilibrium physics to various transport processes/phenomena

Introduction Traffic = macroscopic system of interacting particles Nonequilibrium physics: Driven systems far from equilibrium Various approaches: hydrodynamic gas-kinetic car-following cellular automata

Cellular Automata Cellular automata (CA) are discrete in space time state variable (e.g. occupancy, velocity) Advantage: very efficient implementation for large-scale computer simulations often: stochastic dynamics

Asymmetric Simple Exclusion Process

Asymmetric Simple Exclusion Process (ASEP): 1.directed motion 2.exclusion (1 particle per site) 3.stochastic dynamics q q Caricature of traffic: “Mother of all traffic models” For applications: different modifications necessary

Update scheme random-sequential: site or particles are picked randomly at each step (= standard update for ASEP; continuous time dynamics) parallel (synchronous): all particles or sites are updated at the same time ordered-sequential: update in a fixed order (e.g. from left to right) shuffled: at each timestep all particles are updated in random order In which order are the sites or particles updated ?

ASEP simple exactly solvable many applications Applications: Protein synthesis Surface growth Traffic Boundary induced phase transitions ASEP = “Ising” model of nonequilibrium physics

Periodic boundary conditions no or short-range correlations fundamental diagram

Influence of Boundary Conditions open boundaries: density not conserved! exactly solvable for all parameter values! Derrida, Evans, Hakim, Pasquier 1993 Schütz, Domany 1993

Phase Diagram Low-density phase J=J(p,  ) High-density phase J=J(p,  ) Maximal current phase J=J(p) 1.order transition 2.order transitions

Highway Traffic

Spontaneous Jam Formation Phantom jams, start-stop-waves interesting collective phenomena space time jam velocity: -15 km/h (universal!)

Experiment

Relation: current (flow) $ density Fundamental Diagram free flow congested flow (jams) more detailed features?

Cellular Automata Models Discrete in Space Time State variables (velocity) velocity dynamics: Nagel – Schreckenberg (1992)

Update Rules Rules (Nagel, Schreckenberg 1992) 1)Acceleration: v j ! min (v j + 1, v max ) 2)Braking: v j ! min ( v j, d j ) 3)Randomization: v j ! v j – 1 (with probability p) 4)Motion: x j ! x j + v j (d j = # empty cells in front of car j)

Example Configuration at time t: Acceleration (v max = 2): Braking: Randomization (p = 1/3): Motion (state at time t+1):

Interpretation of the Rules 1)Acceleration: Drivers want to move as fast as possible (or allowed) 2)Braking: no accidents 3)Randomization: a) overreactions at braking b) delayed acceleration c) psychological effects (fluctuations in driving) d) road conditions 4) Driving: Motion of cars

Realistic Parameter Values Standard choice: v max =5, p=0.5 Free velocity: 120 km/h  4.5 cells/timestep Space discretization: 1 cell  7.5 m 1 timestep  1 sec Reasonable: order of reaction time (smallest relevant timescale)

Discrete vs. Continuum Models Simulation of continuum models: Discretisation (  x,  t) of space and time necessary Accurate results:  x,  t ! 0 Cellular automata: discreteness already taken into account in definition of model

Simulation of NaSch Model Reproduces structure of traffic on highways - Fundamental diagram - Spontaneous jam formation Minimal model: all 4 rules are needed Order of rules important Simple as traffic model, but rather complex as stochastic model Simulation

Analytical Methods Mean-field: P(  1,…,  L )¼ P(  1 )  P(  L ) Cluster approximation: P(  1,…,  L )¼ P(  1,  2 ) P(  2,  3 )  P(  L ) Car-oriented mean-field (COMF): P(d 1,…,d L )¼ P(d 1 )  P(d L ) with d j = headway of car j (gap to car ahead) d 1 =1d 2 =0d 3 =2

Particle-hole symmetry Mean-field theory underestimates flow: particle-hole attraction Fundamental Diagram (v max =1) v max =1: NaSch = ASEP with parallel dynamics

ASEP with random-sequential update: no correlations (mean-field exact!) ASEP with parallel update: correlations, mean-field not exact, but 2-cluster approximation and COMF Origin of correlations? Paradisical States Garden of Eden state (GoE) in reduced configuration space without GoE states: Mean-field exact! => correlations in parallel update due to GoE states not true for v max >1 !!! (AS/Schreckenberg 1998) (can not be reached by dynamics!)

Fundamental Diagram (v max >1) No particle-hole symmetry

Phase Transition? Are free-flow and jammed branch in the NaSch model separated by a phase transition? No! Only crossover!! Exception: deterministic limit (p=0) 2nd order transition at

Andreas Schadschneider Institute for Theoretical Physics University of Cologne Germany Modelling of Traffic Flow and Related Transport Problems Lecture II

Nagel-Schreckenberg Model velocity 1.Acceleration 2.Braking 3.Randomization 4.Motion v max =1: NaSch = ASEP with parallel dynamics v max >1: realistic behaviour (spontaneous jams, fundamental diagram)

Fundamental Diagram II free flow congested flow (jams) more detailed features? high-flow states

Metastable States Empirical results: Existence of metastable high-flow states hysteresis

VDR Model Modified NaSch model: VDR model (velocity-dependent randomization) Step 0: determine randomization p=p(v(t)) p 0 if v = 0 p(v) = with p 0 > p p if v > 0 Slow-to-start rule Simulation

NaSch model VDR-model: phase separation Jam stabilized by J out < J max VDR model Jam Structure

Fundamental Diagram III Even more detailed features? non-unique flow- density relation

Synchronized Flow New phase of traffic flow (Kerner – Rehborn 1996) States of high density and relatively large flow velocity smaller than in free flow small variance of velocity (bunching) similar velocities on different lanes (synchronization) time series of flow looks „irregular“ no functional relation between flow and density typically observed close to ramps

3-Phase Theory free flow (wide) jams synchronized traffic 3 phases

Cross-Correlations free flow jam synchro free flow, jam: synchronized traffic: Cross-correlation function: cc  J (  ) / h  (t) J(t+  ) i - h  (t) i h J(t+  )i Objective criterion for classification of traffic phases

Time Headway free flow synchronized traffic many short headways!!! density-dependent

Brake-light model Nagel-Schreckenberg model 1. acceleration (up to maximal velocity) 2. braking (avoidance of accidents) 3. randomization (“dawdle”) 4. motion plus: slow-to-start rule velocity anticipation brake lights interaction horizon smaller cells … Brake-light model (Knospe-Santen-Schadschneider -Schreckenberg 2000) good agreement with single-vehicle data

Fundamental Diagram IV a)Empirical results b)Monte Carlo simulations

Test: „Tunneling of Jams“

Highway Networks Autobahn network of North-Rhine-Westfalia (18 million inhabitants) length: 2500 km 67 intersections (“nodes”) 830 on-/off-ramps (“sources/sinks”)

Data Collection online-data from 3500 inductive loops only main highways are densely equipped with detectors almost no data directly from on-/off-ramps

Online Simulation State of full network through simulation based on available data “interpolation” based on online data: online simulation classification into 4 states (available at

Traffic Forecasting state at 13:51 forecast for 14:56 actual state at 14:54

2-Lane Traffic Rules for lane changes (symmetrical or asymmetrical) Incentive Criterion: Situation on other lane is better Safety Criterion: Avoid accidents due to lane changes

Defects Locally increased randomization: p def > p Ramps have similar effect! shock Defect position

City Traffic BML model: only crossings Even timesteps: " move Odd timesteps: ! move Motion deterministic ! 2 phases: Low densities: hvi > 0 High densities: hvi = 0 Phase transition due to gridlocks

More realistic model Combination of BML and NaSch models Influence of signal periods, Signal strategy (red wave etc), … Chowdhury, Schadschneider 1999

Summary Cellular automata are able to reproduce many aspects of highway traffic (despite their simplicity): Spontaneous jam formation Metastability, hysteresis Existence of 3 phases (novel correlations) Simulations of networks faster than real-time possible Online simulation Forecasting

Finally! Sometimes „spontaneous jam formation“ has a rather simple explanation! Bernd Pfarr, Die ZEIT

Intracellular Transport

Transport in Cells (long-range transport) (short-range transport) microtubule = highway molecular motor (proteins) = trucks ATP = fuel

Molecular Motors DNA, RNA polymerases: move along DNA; duplicate and transcribe DNA into RNA Membrane pumps: transport ions and small molecules across membranes Myosin: work collectively in muscles Kinesin, Dynein: processive enzyms, walk along filaments (directed); important for intracellular transport, cell division, cell locomotion

Microtubule 24 nm 8 nm - +

Mechanism of Motion inchworm: leading and trailing head fixed hand-over-hand: leading and trailing head change Movie

Several motors running on same track simultaneously Size of the cargo >> Size of the motor Collective spatio-temporal organization ? Fuel: ATP ATP ADP + PKinesin Dynein Kinesin and Dynein: Cytoskeletal motors

ASEP-like Model of Molecular Motor-Traffic (Lipowsky, Klumpp, Nieuwenhuizen, 2001 Parmeggiani, Franosch, Frey, 2003 Evans, Juhasz, Santen, 2003)   q DA ASEP + Langmuir-like adsorption-desorption Competition bulk – boundary dynamics

Phase diagram ０ SL H Position of Shock is x=1 when SH x=0 when LS 01 L H S cf. ASEP

General belief: Coordination of two heads is required for processivity (i.e., long-distance travel along the track) of conventional TWO-headed kinesin. KIF1A is a single-headed processive motor. Then, why is single-headed KIF1A processive? Single-headed kinesin KIF1A Movie

2-State Model for KIF1A state 1: “strongly bound” state 2: “weakly bound” Hydrolysis cycle of KIF1A KKT KDPKD ATP P ADP Bound on MT Brownian & Ratchet motion on MT hydrolysis 1 2

New model for KIF1A ー＋ Brownian, ratchet AttachmentDetachment t t Brownian Release ADP （ Ratchet ） Hydrolysis Att.Det.

0.01 (0.0094) 0.1 (0.15) (1) (100) 0.2 (0.9) Blue: state_1 Red: state_ (5) Phase diagram

position of domain wall can be measured as a function of controllable parameters. Nishinari, Okada, Schadschneider, Chowdhury, Phys. Rev. Lett. (2005) KIF1A (Red) MT (Green) 10 pM 100 pM 1000pM 2 mM of ATP 2  m Spatial organization of KIF1A motors: experiment

Andreas Schadschneider Institute for Theoretical Physics University of Cologne Germany Modelling of Traffic Flow and Related Transport Problems Lecture III

Dynamics on Ant Trails

Ant trails ants build “road” networks: trail system

Chemotaxis Ants can communicate on a chemical basis: chemotaxis Ants create a chemical trace of pheromones trace can be “smelled” by other ants follow trace to food source etc.

Chemotaxis chemical trace: pheromones

Ant trail model Basic ant trail model: ASEP + pheromone dynamics hopping probability depends on density of pheromones distinguish only presence/absence of pheromones ants create pheromones ‘free’ pheromones evaporate

q q Q 1.motion of ants 2.pheromone update (creation + evaporation) Dynamics: f f f parameters: q < Q, f Ant trail model qqQ equivalent to bus-route model (O’Loan, Evans Cates 1998) (Chowdhury, Guttal, Nishinari, A.S. 2002)

Limiting cases f=0: pheromones never evaporate => hopping rate always Q in stationary state f=1: pheromone evaporates immediately => hopping rate always q in stationary state for f=0 and f=1: ant trail model = ASEP (with Q, q, resp.)

Fundamental diagram of ant trails different from highway traffic: no egoism velocity vs. density Experiments: Burd et al. (2002, 2005) non-monotonicity at small evaporation rates!!

Experimental result (Burd et al., 2002) Problem: mixture of unidirectional and counterflow

Spatio-temporal organization formation of “loose clusters” early timessteady state coarsening dynamics: cluster velocity ~ gap to preceding cluster

Traffic on Ant Trails Formation of clusters

Analytical Description Mapping on Zero-Range Process ant trail model: (v = average velocity) phase transition for f ! 0 at

Counterflow hindrance effect through interactions (e.g. for communication) plateau

Pedestrian Dynamics

Collective Effects jamming/clogging at exits lane formation flow oscillations at bottlenecks structures in intersecting flows

Lane Formation

Oscillations of Flow Direction

Pedestrian Dynamics More complex than highway traffic motion is 2-dimensional counterflow interaction “longer-ranged” (not only nearest neighbours)

Pedestrian model Modifications of ant trail model necessary since motion 2-dimensional: diffusion of pheromones strength of trace idea: Virtual chemotaxis chemical trace: long-ranged interactions are translated into local interactions with ‘‘memory“

Long-ranged Interactions Problems for complex geometries: Walls ’’screen“ interactions Models with local interactions ???

Floor field cellular automaton Floor field CA: stochastic model, defined by transition probabilities, only local interactions reproduces known collective effects (e.g. lane formation) Interaction: virtual chemotaxis (not measurable!) dynamic + static floor fields interaction with pedestrians and infrastructure

Static Floor Field Not influenced by pedestrians no dynamics (constant in time) modelling of influence of infrastructure Example: Ballroom with one exit

Transition Probabilities Stochastic motion, defined by transition probabilities 3 contributions: Desired direction of motion Reaction to motion of other pedestrians Reaction to geometry (walls, exits etc.) Unified description of these 3 components

Transition Probabilities Total transition probability p ij in direction (i,j): p ij = N¢ M ij exp(k D D ij ) exp(k S S ij )(1-n ij ) M ij = matrix of preferences (preferred direction) D ij = dynamic floor field (interaction between pedestrians) S ij = static floor field (interaction with geometry) k D, k S = coupling strength N = normalization (  p ij = 1)

Lane Formation velocity profile

Friction Friction: not all conflicts are resolved! (Kirchner, Nishinari, Schadschneider 2003) friction constant  = probability that no one moves Conflict: 2 or more pedestrians choose the same target cell

Herding Behaviour vs. Individualism Minimal evacuation times for optimal combination of herding and individual behaviour Evacuation time as function of coupling strength to dynamical floor field (Kirchner, Schadschneider 2002) Large k D : strong herding

Evacuation Scenario With Friction Effects Faster-is-slower effect evacuation time effective velocity (Kirchner, Nishinari, A.S. 2003)

Competitive vs. Cooperative Behaviour Experiment: egress from aircraft (Muir et al. 1996) Evacuation times as function of 2 parameters: motivation level - competitive (T comp ) - cooperative (T coop ) exit width w

Empirical Egress Times T comp > T coop for w < w c T comp w c

Model Approach Competitive behaviour: large k S + large friction  Cooperative behaviour: small k S + no friction  =0 (Kirchner, Klüpfel, Nishinari, A. S., Schreckenberg 2003)

Summary Variants of the Asymmetric Simple Exclusion Process Highway traffic: larger velocities Ant trails: state-dependent hopping rates Pedestrian dynamics: 2-dimensional motion Intracellular transport: adsorption + desorption Various very different transport and traffic problems can be described by similar models

Applications Highway traffic: Traffic forecasting Traffic planning and optimization Ant trails: Optimization of traffic Pedestrian dynamics (virtual chemotaxis) Pedestrian dynamics: safety analysis (planes, ships, football stadiums,…) Intracellular transport: relation with diseases (ALS, Alzheimer,…)

Collaborators Cologne: Ludger Santen Ansgar Kirchner Alireza Namazi Kai Klauck Frank Zielen Carsten Burstedde Alexander John Philip Greulich Thanx to: Rest of the world: Debashish Chowdhury (Kanpur) Ambarish Kunwar (Kanpur) Vishwesha Guttal (Kanpur) Katsuhiro Nishinari (Tokyo) Yasushi Okada (Tokyo) Gunter Schütz (Jülich) Vladislav Popkov (now Cologne) Kai Nagel (Berlin) Janos Kertesz (Budapest) Duisburg: Michael Schreckenberg Robert Barlovic Wolfgang Knospe Hubert Klüpfel Torsten Huisinga Andreas Pottmeier Lutz Neubert Bernd Eisenblätter Marko Woelki